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Notre Dame 2015

Explicit realization of affine vertex algebras and their applications

Draˇ zen Adamovi´ c University of Zagreb, Croatia Supported by CSF, grant. no. 2634 Conference on Lie algebras, vertex operator algebras and related topics A conference in honor of J. Lepowsky and R. Wilson University of Notre Dame August 14 - 18, 2015

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Affine vertex algebras

Let V k(g) be universal affine vertex algebra of level k associated to the affine Lie algebra ˆ g. V k(g) is generated generated by the fields x(z) =

n∈Z x(n)z−n−1,

x ∈ g. As a ˆ g–module, V k(g) can be realized as a generalized Verma module. For every k ∈ C, the irreducible ˆ g–module Lk(g) carries the structure of a simple vertex algebra.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Affine Lie algebra A(1)

1 Let now g = sl2(C) with generators e, f , h and relations [h, e] = 2e, [h, f ] = −2f , [e, f ] = h. The corresponding affine Lie algebra ˆ g is of type A(1)

1 .

The level k = −2 is called critical level. For x ∈ sl2 identify x with x(−1)1. Let Θ be the automorphism of V k(sl2) such that Θ(e) = f , Θ(f ) = e, Θ(h) = −h.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Affine Lie algebra A(1)

1

in principal graduation

Let sl2[Θ] be the affine Lie algebra sl2 in principal graduation [Lepowsky-Wilson]. sl2[Θ] has basis: {K, h(m), x+(n), x−(p) |m, p ∈ 1

2 + Z, n ∈ Z}

with commutation relations:

[h(m), h(n)] = 2mδm+n,0K [h(m), x+(r)] = 2x−(m + r) [h(m), x−(n)] = 2x+(m + n) [x+(r), x+(s)] = 2rδr+s,0K [x+(r), x−(m)] = −2h(m + r) [x−(m), x−(n)] = −2mδm+n,0K K in the center

• Proposition. (FLM)

The category of Θ–twisted V k(sl2)–modules coincides with the category

• f restricted modules for

sl2[Θ] of level k.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

N = 2 superconformal algebra

N = 2 superconformal algebra (SCA) is the infinite-dimensional Lie superalgebra with basis L(n), H(n), G±(r), C, n ∈ Z, r ∈ 1

2 + Z and

(anti)commutation relations given by

[L(m), L(n)] = (m − n)L(m + n) + C

12(m3 − m)δm+n,0,

[H(m), H(n)] = C

3 mδm+n,0,

[L(m), G±(r)] = ( 1

2m − r)G±(m + r),

[L(m), H(n)] = −nH(n + m), [H(m), G±(r)] = ±G±(m + r), {G+(r), G−(s)} = 2L(r + s) + (r − s)H(r + s) + C

3 (r 2 − 1 4)δr+s,0,

[L(m), C] = [H(n), C] = [G±(r), C] = 0, {G+(r), G+(s)} = {G−(r), G−(s)} = 0

for all m, n ∈ Z, r, s ∈ 1

2 + Z.

Let V N=2

c

be the universal N = 2 superconformal vertex algebra.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

N = 2 superconformal algebra

N = 2 superconformal algebra (SCA) admits the mirror map automorphism (terminology of K. Barron): κ : G±(r) → G∓(r), H(m) → −H(m), L(m) → L(m), C → C which can be lifted to an automorphism of V N=2

c

.

• Proposition. (K. Barron, ..)

The category of κ–twisted V N=2

c

–modules coincides with the category of restricted modules for the mirror twisted N = 2 superconformal algebra

• f central charge c.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Correspondence

When k = −2, the representation theory of the affine Lie algebra A(1)

1

is related with the representation theory of the N = 2 superconformal algebra. The correspondence is given by Kazama-Suzuki mappings. We shall extend this correspondence to representations at the critical level by introducing a new infinite-dimensional Lie superalgebra A.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Vertex superalgebras F and F−1

The Clifford vertex superalgebra F is generated by fields Ψ±(z) =

n∈Z Ψ±(n + 1 2)z−n−1,

whose components satisfy the (anti)commutation relations for the infinite dimensional Clifford algebra CL:

{Ψ±(r), Ψ∓(s)} = δr+s,0; {Ψ±(r), Ψ±(s)} = 0 (r, s ∈ 1

2 + Z).

Let F−1 = M(1) ⊗ C[L] be the lattice vertex superalgebra associated to the lattice L = Zβ, β, β = −1.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Vertex superalgebras F and F−1

Let ΘF be automorphism of order two of F lifted from the automorphism Ψ±(r) → Ψ∓(r) of the Clifford algebra. F has two inequivalent irreducible ΘF–twisted modules F Ti. Let ΘF−1 be the automorphism of F−1–lifted from the automorphism β → −β of the lattice L. F−1 has two inequivalent irreducible ΘF−1–twisted modules F Ti

−1

realized on MZ+ 1

2

(1) = C[β(− 1

2), β(− 3 2), . . . ].

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

N = 2 superconformal vertex algebra

Let g = sl2. Consider the vertex superalgebra V k(g) ⊗ F. Define τ + = e(−1) ⊗ Ψ+(− 1

2),

τ − = f (−1) ⊗ Ψ−(− 1

2).

Then the vertex subalgebra of V k(g) ⊗ F generated by τ + and τ − carries the structure of a highest weight module for of the N = 2 SCA: G±(z) =

• 2

k+2Y (τ ±, z) = n∈Z G±(n + 1 2)z−n−2

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Kazama-Suzuki and ”anti” Kazama-Suzuki mappings

Introduced by Fegin, Semikhatov and Tipunin (1997) Assume that M is a (weak ) V k(g)-module. Then M ⊗ F is a (weak) V N=2

c

–module with c = 3k/(k + 2). Assume that N is a weak V N=2

c

–module. Then N ⊗ F−1 is a (weak ) V k(sl2)–module. This enables a classification of irreducible modules for simple vertex superalgebras associated to N=2 SCA (D.Adamovi´ c, IMRN (1998) )

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Kazama-Suzuki and ”anti” Kazama-Suzuki mappings: twisted version

Let c = 3k/(k + 2). Assume that Mtw is a Θ–twisted V k(g)-module. Then Mtw ⊗ F Ti is a κ–twisted V N=2

c

–module. Assume that Ntw is a κ–twisted V N=2

c

–module. Then Ntw ⊗ F Ti

−1 is

a Θ–twisted V k(g)–module.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Lie superalgebra A

A is infinite-dimensional Lie superalgebra with generators S(n), T(n), G ±(r), C, n ∈ Z, r ∈ 1

2 + Z, which satisfy the following

relations S(n), T(n), C are in the center of A, {G +(r), G −(s)} = 2S(r + s) + (r − s)T(r + s) + C

3 (r 2 − 1 4)δr+s,0,

{G +(r), G +(s)} = {G −(r), G −(s)} = 0 for all n ∈ Z, r, s ∈ 1

2 + Z.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

The (universal) vertex algebra V

V is strongly generated by the fields G ±(z) = Y (τ ±, z) =

• n∈Z

G ±(n + 1

2)z−n−2,

S(z) = Y (ν, z) =

• n∈Z

S(n)z−n−2, T(z) = Y (j, z) =

• n∈Z

T(n)z−n−1. The components of these fields satisfy the (anti)commutation relations for the Lie superalgebra A. Let ΘV be the automorphism of V lifted from the automorphism of

• rder two of A such that

G ±(r) → G ∓(r), T(r) → −T(r), S(r) → S(r), C → C.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Lie superalgebra Atw

Atw has the basis S(n), T (n + 1/2), G(r), C, n ∈ Z, r ∈ 1

2Z

and anti-commutation relations: {G(r), G(s)} = (−1)2r+1(2δZ

r+sS(r+s)−δ 1 2 +Z r+s (r−s)Tr+s+ C 3 δZ r+s(r 2− 1 4) δr+s,0),

S(n), T (n + 1/2), C in the center, with δS

m = 1 if m ∈ S, δS m = 0 otherwise.

Proposition. The category of ΘV–twisted V–modules coincides with the category of restricted modules for the Lie superalgebra Atw.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Theorem (A, CMP 2007) Assume that U is an irreducible V–module such that U admits the following Z–gradation U =

• j∈Z

Uj, Vi.Uj ⊂ Ui+j. Let F−1 be the vertex superalgebra associated to lattice Z√−1. Then U ⊗ F−1 =

• s∈Z

Ls(U), where Ls(U) :=

• i∈Z

Ui ⊗ F −s+i

−1

and for every s ∈ Z Ls(U) is an irreducible A(1)

1 –module at the critical

level.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Weyl vertex algebra

The Weyl vertex algebra W is generated by the fields a(z) =

• n∈Z

a(n)z−n−1, a∗(z) =

• n∈Z

a∗(n)z−n, whose components satisfy the commutation relations for infinite-dimensional Weyl algebra [a(n), a(m)] = [a∗(n), a∗(m)] = 0, [a(n), a∗(m)] = δn+m,0.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Wakimoto modules

Assume that χ(z) ∈ C((z)). On the vertex algebra W exists the structure of the A(1)

1 –module at

the critical level defined by e(z) = a(z), h(z) = −2 : a∗(z)a(z) : −χ(z) f (z) = − : a∗(z)2a(z) : −2∂za∗(z) − a∗(z)χ(z). This module is called the Wakimoto module and it is denoted by W−χ(z).

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Theorem (D.A., CMP 2007, Contemp. Math. 2014) The Wakimoto module W−χ is irreducible if and only if χ(z) satisfies one

• f the following conditions:

(i) There is p ∈ Z>0, p ≥ 1 such that χ(z) =

∞

• n=−p

χ−nzn−1 ∈ C((z)) and χp = 0. (ii) χ(z) = ∞

n=0 χ−nzn−1 ∈ C((z))

and χ0 ∈ {1} ∪ (C \ Z). (iii) There is ℓ ∈ Z≥0 such that χ(z) = ℓ + 1 z +

∞

• n=1

χ−nzn−1 ∈ C((z)) and Sℓ(−χ) = 0, where Sℓ(−χ) = Sℓ(−χ−1, −χ−2, . . . ) is a Schur polynomial.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

The structure of Wakimto modules

Theorem Assume that χ ∈ C((z)) such that Wχ is reducible. Then (1) Wχ is indecomposable. (2) The maximal sl2–integrable submodule W int

χ

is irreducible. (3) Wχ/W int

χ

is irreducible.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Theorem (D. Adamovic, N. Jing, K. Misra, 2014-2015) Assume that Utw is an irreducible, restricted Atw–module, and F Ti

−1

twisted F−1–module. Then Utw ⊗ F Ti

−1 has the structure of irreducible

• sl2[Θ]–module at the critical level.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Construction of Atw–modules

Let F Ti irreducible ΘF–twisted F–module. FTi is an irreducible module the the twisted Clifford algebra CLtw with generators Φ(r), r ∈ 1

2Z, and relations

{Φ(r), Φ(s)} = −(−1)2rδr+s,0; (r, s ∈ 1 2Z) Let χ ∈ C((z1/2)). The Atw–module F Ti(χ) is uniquely determined by G(z) = ∂zΦ(z) + χ(z)Φ(z) =

• n∈ 1

2 Z

G(n)z−n−1.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Theorem (D. Adamovic, N. Jing, K. Misra, 2014-2015) Assume that p ∈ 1

2Z>0 and that

χ(z) =

∞

• k=−2p

χ− k

2

z

k 2 −1.

Then F Ti(χ) is irreducible Atw–module if and only if one of the following conditions hold: p > 0 and χp = 0, (1) p = 0 and χ0 ∈ (C \ 1

2Z) ∪ {1

2}, (2) p = 0 and χ0 − 1 2 = ℓ ∈ 1

2Z>0

and det(A(χ)) = 0 (3)

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015 A(χ) =           2S(−1) 2S(−2) · · · 2S(−2ℓ) ℓ2 − (ℓ − 1)2 2S(−1) 2S(−2) · · · 2S(−2ℓ − 1) ℓ2 − (ℓ − 2)2 2S(−1) · · · 2S(−2ℓ − 2) . . . . . . . . . . . . . . . · · · ℓ2 − (ℓ − 2ℓ + 1)2 2S(−1)          

and S(z) = 1 2(χ(1)(z))2 + ∂zχ(1)(z)) =

• n∈Z

S(n)z−n−2 (here χ(1) is the integral part of χ).

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

N=4 superconformal vertex algebra V N=4

c V N=4

c

is generated by the Virasoro field L, three primary fields of conformal weight 1, J0, J+ and J− (even part) and four primary fields of conformal weight 3

2, G ± and G ± (odd part).

The remaining (non-vanishing) λ–brackets are [J0

λ, J±] = ±2J±

[J0

λJ0] = c

3l [J+

λ J−] = J0 + c 6λ

[J0

λG ±] = ±G ±

[J0

λG ±] = ±G ±

[J+

λ G −] = G +

[J−

λ G +] = G −

[J+

λ G −] = −G +

[J−

λ G +] = −G −

[G ±

λ G ±] = (T + 2λ)J±

[G ±

λ G ∓] =

L ± 1 2TJ0 ± λJ0 + c 6λ2 Let LN=4

c

be its simple quotient.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

N=4 superconformal vertex algebra LN=4

c

with c = −9

We shall present some results from D.Adamovi´ c, arXiv:1407.1527. (to appear in Transformation Groups) Theorem (i) The simple affine vertex algebra Lk(sl2) with k = −3/2 is conformally embedded into LN=4

c

with c = −9. (ii) LN=4

c

∼ = (M ⊗ F)int where M ⊗ F is a maximal sl2–integrable submodule of the Weyl-Clifford vertex algebra M ⊗ F.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

LN=4

c

with c = −9 as an sl2–module

LN=4

c

with c = −9 is completely reducible sl2–module and the following decomposition holds: LN=4

c

∼ =

∞

• m=0

(m + 1)LA1(−(3 2 + n)Λ0 + nΛ1). LN=4

c

is a completely reducible sl2 × sl2–modules. sl2 action is

• btained using screening operators for Wakimoto realization of
• sl2–modules at level −3/2.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

The affine vertex algebra Lk(sl3) with k = −3/2.

Theorem (i) The simple affine vertex algebra Lk(sl3) with k = −3/2 is realized as a subalgebra of LN=4

c

⊗ F−1 with c = −9. In particular Lk(sl3) can be realized as subalgebra of M ⊗ F ⊗ F−1. (ii) LN=4

c

⊗ F−1 is a completely reducible A(1)

2 –module at level k = −3/2.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

On representation theory of LN=4

c

with c = −9

LN=4

c

has only one irreducible module in the category of strong

• modules. Every Z>0–graded LN=4

c

–module with finite-dimensional weight spaces (with respect to L(0)) is semisimple (”Rationality in the category of strong modules”) LN=4

c

has two irreducible module in the category O. There are non-semisimple LN=4

c

–modules from the category O. LN=4

c

has infinitely many irreducible modules in the category of weight modules. LN=4

c

admits logarithmic modules on which L(0) does not act semi-simply.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Theorem (D.A, 2014) Assume that U is an irreducible LN=4

c

–module with c = −9 such that U =

j∈Z Uj is Z–graded (in a suitable sense).

Let F−1 be the vertex superalgebra associated to lattice Z√−1. Then U ⊗ F−1 =

• s∈Z

Ls(U), where Ls(U) :=

• i∈Z

Ui ⊗ F −s+i

−1

and for every s ∈ Z Ls(U) is an irreducible A(1)

2 –module at level −3/2.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Connection with C2–cofinite vertex algebras appearing in LCFT

Drinfeld-Sokolov reduction maps: LN=4

c

to doublet vertex algebra A(p) and even part (LN=4

c

)even to triplet vertex algebra W(p) with p = 2 (symplectic-fermion case) Vacuum space of Lk(sl3) with k = −3/2 contains the vertex algebra WA2(p) with p = 2 (which is conjecturally C2–cofinite).

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Connection with C2–cofinite vertex algebras appearing in LCFT:

Vacuum space of Lk(sl3) with k = −3/2 contains the vertex algebra WA2(p) with p = 2 (which is conjecturally C2–cofinite). Affine vertex algebra Lk(sl2) for k + 2 = 1

p, p ≥ 2 can be

conformally embedded into the vertex algebra V(p) generated by Lk(sl2) and 4 primary vectors τ ±

(p), τ ± (p).

V(p) ∼ = LN=4

c

for p = 2. Drinfeld-Sokolov reduction maps V(p) to the doublet vertex algebra A(p) and even part (V(p))even to the triplet vertex algebra W(p). (C2–cofiniteness and RT of these vertex algebras were obtain in a work of D.A and A. Milas)

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

The Vertex algebra WA2(p): Definition

We consider the lattice √pA2 = Zγ1 + Zγ2, γ1, γ1 = γ2, γ2 = 2p, γ1, γ2 = −p. Let Mγ1,γ2(1) be the s Heisenberg vertex subalgebra of V√pA2 generated by the Heisenberg fields γ1(z) and γ2(z). WA2(p) = KerV√pA2 e−γ1/p

• KerV√pA2 e−γ2/p

. We also have its subalgebra: W0

A2(p) = KerMγ1,γ2(1)e−γ1/p

• KerMγ1,γ2(1)e−γ2/p

WA2(p) and W0

A2(p) have vertex subalgebra isomorphic to the simple

W(2, 3)–algebra with central charge cp = 2 − 24 (p−1)2

p

.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

The Vertex algebra WA2(p): Conjecture

(i) WA2(p) is a C2–cofinite vertex algebra for p ≥ 2 and that it is a completely reducible W(2, 3) × sl3–module. (ii) WA2(p) is strongly generated by W(2, 3) generators and by sl3.e−γ1−γ2, so by 8 primary fields for the W(2, 3)–algebra. Note that WA2(p) is a generalization of the triplet vertex algebra W(p) and W0

A2(p) is a generalization of the singlet vertex

subalgebra of W(p).

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Relation with parafermionic vertex algebras for p = 2

(i) Let K(sl3, k) be the parafermion vertex subalgebra of Lk(sl3) (C. Dong talk). (iii) For k = −3/2 we have K(sl3, k) = W0

A2(p).

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

General conjectures

Conjecture Assume that Lk(g) is a simple affine vertex algebra of affine type and k ∈ (Q \ Z≥0) is admissible. Then The vacuum space Ω(Lk(g)) = {v ∈ Lk(g) | h(n).v = 0 h ∈ h, n ≥ 1} is extension of certain C2–cofinite, irrational vertex algebra. Lk(g) admits logarithmic representations. (1) Ω(Lk(sl2) ∼ = W (2) for k = −1/2. (2) Ω(Lk(sl2) ∼ = A(3) for k = −4/3, where A(3) is SCE extension of triplet vertex algebra W (3). (3) Ω(Lk(sp2n)) for k = −1/2 is Z2–orbifold of symplectic fermions.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Realization of simple W –algebras

Let F−p/2 denotes the generalized lattice vertex algebra associated to the lattice Z( p

2ϕ) such that

ϕ, ϕ = −2 p . Let R(p) by the subalgebra of V(p) ⊗ F−p/2 generated by x = x(−1)1 ⊗ 1, x ∈ {e, f , h}, 1 ⊗ ϕ(−1)1 and eα1,p := 1 √ 2 τ +

(p) ⊗ e p 2 ϕ

(4) fα1,p := 1 √ 2 τ −

(p) ⊗ e− p 2 ϕ

(5) eα2,p := 1 √ 2 τ +

(p) ⊗ e− p 2 ϕ

(6) fα2,p := 1 √ 2 τ −

(p) ⊗ e p 2 ϕ

(7)

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Realization of simple W –algebras

R(2) ∼ = LA2(− 3

2Λ0).

R(3) ∼ = Wk(sl4, fθ) with k = −8/3. (Conjecture) R(p) and V(p) have finitely many irreducible modules in the category O. R(p) and V(p) have infinitely many irreducible modules outside of the category O and admit logarithmic modules.

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications

Notre Dame 2015

Thank you

Draˇ zen Adamovi´ c Explicit realization of affine vertex algebras and their applications